Efron-Stein inequalities for random matrices (Q341495)

The authors' abstract: ``This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron-Stein inequalities developed by \textit{S. Boucheron} et al. [ibid. 31, No. 3, 1583--1614 (2003; Zbl 1051.60020); ibid. 33, No. 2, 514--560 (2005; Zbl 1074.60018)]. The proofs rely on the method of exchangeable pairs.''NEWLINENEWLINE The paper is very large structured in 13 chapters with subchapters and an appendix with three parts:NEWLINENEWLINE 1. Introduction (and technical approach).NEWLINENEWLINE 2. Notation and preliminaries from matrix analysis -- 2.1 Elementary matrices, 2.2 Sets of matrices and the semidefinite order, 2.3 Matrix functions, 2.4 Monotonicity and convexity of trace functions, 2.5 The real part of a matrix and the matrix square, 2.6 Some matrix norms.NEWLINENEWLINE 3. Matrix moments and concentration -- 3.1 The matrix Chebyshev inequality, 3.2 The matrix Laplace transform method.NEWLINENEWLINE 4. Matrix Efron-Stein inequalities -- 4.1 Setup for Efron-Stein inequalities, 4.2 Polynomial Efron-Stein inequalities for random matrices, 4.3 Exponential Efron-Stein inequalities for random matrices, 4.4 Rectangular matrices.NEWLINENEWLINE 5. Example: Self-bounded random matrices.NEWLINENEWLINE 6. Example: Matrix bounded differences.NEWLINENEWLINE 7. Application: Compound sample covariance matrices -- 7.1 Setup, 7.2 A bound for the variance proxy, 7.3 A bound for the trace m.g.f. of the random matrix, 7.4 The positive-semidefinite case, 7.5 The general case.NEWLINENEWLINE 8. Random matrices, exchangeable pairs and kernels -- 8.1 Exchangeable pairs, 8.2 Kernel Stein pairs, 8.3 The method of exchangeable pairs, 8.4 Conditional variances.NEWLINENEWLINE 9. Polynomial moments of a random matrix -- 9.1 The polynomial mean value trace inequality, 9.2 Proof of Theorem 9.1.NEWLINENEWLINE 10. Constructing a kernel via Markov chain coupling -- 10.1 Overview, 10.2 Kernel couplings, 10.3 Kernel Stein pairs from the Poisson equation, 10.4 Bounding the conditional variances I, 10.5 Bounding the conditional variances II.NEWLINENEWLINE 11. The polynomial Efron-Stein inequality for a random matrix -- 11.1 A kernel coupling for a vector of independent variables, 11.2 A kernel Stein pair, 11.3 The evolution of the kernel coupling, 11.4 Conditional variance bounds, 11.5 The polynomial Efron-Stein inequality: Bounded case, 11.6 The polynomial Efron-Stein inequality: General case.NEWLINENEWLINE 12. Exponential concentration inequalities -- 12.1 Proof of exponential Efron-Stein inequality: Bounded case, 12.2 The exponential Efron-Stein inequality: General case, 12.3 The exponential mean value trace inequality, 12.4 Some properties of the trace m.g.f., 12.5 Bounding the derivative of the trace m.g.f., 12.6 Decoupling via an entropy inequality, 12.7 A differential inequality, 12.8 Solving the differential inequality.NEWLINENEWLINE 13. Complements -- 13.1 Matrix bounded differences without independence, 13.2 Matrix-valued functions of Haar random elements, 13.3 Conjectures and consequences.NEWLINENEWLINE Appendix A: Operator inequalities -- A.1 Young's inequality for commuting operators, A.2 An operator version of Cauchy-Schwarz.NEWLINENEWLINE Appendix B: The polynomial mean value trace inequality.NEWLINENEWLINE Appendix C: The exponential mean value trace inequality.NEWLINENEWLINE The paper contains 40 references on this topic.NEWLINENEWLINE Every chapter has an own numbering for the remarks, lemmas, theorems, corollaries, definitions and examples, where most of them have in brackets short comments. This is helpful for a good understanding of the following texts, e.g. Definition 3.2 (Trace m.g.f.), Proposition 3.3 (Matrix Laplace transform method), Theorem 4.2 (Matrix polynomial Efron-Stein), Theorem 4.3 (Matrix exponential Efron-Stein), Theorem 7.1 (Concentration of compound sample covariance), Definition 8.1 (Exchangeable pair).